Optimal. Leaf size=881 \[ -\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{2 \sqrt [4]{c x^2+b x+a}}{e} \]
[Out]
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Rubi [A] time = 2.63761, antiderivative size = 881, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {734, 843, 623, 220, 749, 748, 747, 401, 108, 409, 1213, 537, 444, 63, 212, 208, 205} \[ -\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{c^{3/4} e^{3/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{2 \sqrt [4]{c x^2+b x+a}}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 734
Rule 843
Rule 623
Rule 220
Rule 749
Rule 748
Rule 747
Rule 401
Rule 108
Rule 409
Rule 1213
Rule 537
Rule 444
Rule 63
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x+c x^2}}{d+e x} \, dx &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\int \frac{b d-2 a e+(2 c d-b e) x}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{2 e}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{(2 c d-b e) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{2 e^2}-\frac{(e (b d-2 a e)-d (2 c d-b e)) \int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{3/4}} \, dx}{2 e^2}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\left (2 (2 c d-b e) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{e^2 (b+2 c x)}-\frac{\left ((e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \int \frac{1}{(d+e x) \left (-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}\right )^{3/4}} \, dx}{2 e^2 \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (\sqrt{2} (e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{c (2 c d-b e)}{b^2-4 a c}+e x\right ) \left (1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4}} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{e^2 \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}+\frac{\left (\sqrt{2} (e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{e \left (a+b x+c x^2\right )^{3/4}}+\frac{\left (\sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\left (b^2-4 a c\right ) x^2}{c^2}\right )^{3/4} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) e^2 \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}+\frac{\left ((e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2\right )}{\sqrt{2} e \left (a+b x+c x^2\right )^{3/4}}+\frac{\left (c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{\left (b^2-4 a c\right ) x}{c^2}} \left (1-\frac{\left (b^2-4 a c\right ) x}{c^2}\right )^{3/4} \left (\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2\right )}{\sqrt{2} \left (b^2-4 a c\right ) e^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (2 \sqrt{2} c^2 (e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c^2 e^2}{b^2-4 a c}+\frac{c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}+\frac{c^2 e^2 x^4}{b^2-4 a c}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) e \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (2 \sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4} \left (-e^2+\frac{(2 c d-b e)^2}{b^2-4 a c}+e^2 x^4\right )} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) e^2 \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (\left (b^2-4 a c\right ) (e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}-\sqrt{-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (\left (b^2-4 a c\right ) (e (b d-2 a e)-d (2 c d-b e)) \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}+\sqrt{-b^2+4 a c} e x^2} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{c} e \sqrt{c d^2-b d e+a e^2} \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (\sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) e^2 \left (-e^2+\frac{(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (\sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right ) \sqrt{1-x^4}} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) e^2 \left (-e^2+\frac{(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (\sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (1-\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) e^2 \left (-e^2+\frac{(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (\sqrt{2} c (2 c d-b e) (e (b d-2 a e)-d (2 c d-b e)) \sqrt{\frac{\left (b^2-4 a c\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (1+\frac{\sqrt{-b^2+4 a c} e x^2}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}}\right )} \, dx,x,\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) e^2 \left (-e^2+\frac{(2 c d-b e)^2}{b^2-4 a c}\right ) \left (-\frac{b c}{b^2-4 a c}-\frac{2 c^2 x}{b^2-4 a c}\right ) \left (a+b x+c x^2\right )^{3/4}}\\ &=\frac{2 \sqrt [4]{a+b x+c x^2}}{e}-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (-b^2+4 a c\right )^{3/4} \sqrt [4]{c d^2-b d e+a e^2} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{-b^2+4 a c} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{c^{3/4} e^{3/2} \left (a+b x+c x^2\right )^{3/4}}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt [4]{c} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (-\frac{\sqrt{-b^2+4 a c} e}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{3/4} \Pi \left (\frac{\sqrt{-b^2+4 a c} e}{2 \sqrt{c} \sqrt{c d^2-b d e+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} c e^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 2.27345, size = 617, normalized size = 0.7 \[ \frac{\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} (b e-2 c d) \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )}{c}+\frac{\left (4 a c-b^2\right )^{3/4} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \left (-\sqrt{2} \sqrt [4]{c} \sqrt{e} (b+2 c x) \sqrt [4]{e (a e-b d)+c d^2} \left (\tan ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )+\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt [4]{4 a c-b^2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}}{\sqrt [4]{c} \sqrt [4]{e (a e-b d)+c d^2}}\right )\right )+\sqrt [4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left .-\sin ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )+\sqrt [4]{4 a c-b^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} (b e-2 c d) \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2+e (a e-b d)}};\left .-\sin ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}}\right )\right |-1\right )\right )}{\sqrt{2} c (b+2 c x)}+2 e (a+x (b+c x))}{e^2 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.296, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}\sqrt [4]{c{x}^{2}+bx+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{a + b x + c x^{2}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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